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z = (1 + i)^2/(a - i) with a > 0 has a given modulus: find z or its conjugate

The same question, asked 3 times: 2 May 2023 S1*, 12 May 2024 S1*, 14 May 2024 S1. Every wording below, oldest first.

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  1. Question 1

    2 May 2023 S1

    118
    If a>0a>0 and z=(1+i)2a−iz=\dfrac{(1+i)^2}{a-i}, i=−1i=\sqrt{-1}, has magnitude 25\sqrt{\dfrac{2}{5}}, then zˉ\bar{z} is equal to

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  2. Question 2

    12 May 2024 S1

    124
    If a>0a>0 and z=(1+i)2a+iz=\frac{(1+i)^{2}}{a+i} (i=−1i=\sqrt{-1}) has magnitude 25\frac{2}{\sqrt{5}}, then zˉ\bar{z} is equal to

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  3. Question 3

    14 May 2024 S1

    136
    If a>0a>0 and z=(1+i)2a−iz=\dfrac{(1+i)^2}{a-i}, i=−1i=\sqrt{-1}, has magnitude 25\dfrac{2}{\sqrt{5}}, then zz is

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